The stalemate means that Francisco Rodriguez's team will end the campaign one point above the drop zone.
Chances were few and far between during a nervy first half, but Almeria almost made the breakthrough when Jonathan Zongo shot narrowly off target.
Bilbao's best opportunity of the opening 45 minutes came late on as Aymeric Laporte saw his effort go inches wide of the post following a corner.
The visitors were dominant after the break as they continued to pin their opponents back, yet it was Almeria who nearly took the lead against the run of play. However, Alex Vidal's low drive was pushed away by goalkeeper Iago Herrerin on 63 minutes.
Almeria came close again in the final five minutes as Rodri Rios was denied by Herrerin for a second time before shooting another effort wide.